Measuring power consumption efficiency of an electromechanical system within a long-term period by fuzzy DEA and TOPSIS for sustainability

Abstract

To realize and handle power consumption details of sustainability, power consumption efficiency measurement for an electromechanical system within a long-term period is critical. Practically, data envelopment analysis (DEA) is useful to measure power consumption efficiency of an electromechanical system for yielding relative efficiencies of peer decision-making units (DMUs). Generally, traditional DEA scarcely considered undesirable outputs, and thus Hwang et al. developed DEA with linear programming to measure DMUs with undesirable outputs. Based on above, inputs, desirable outputs, and undesirable outputs of DMUs in this paper are assumed into triangular fuzzy numbers for grasping data uncertainty and obtaining more messages. Since inputs and outputs of DMUs in Hwang et al.’s were shown by crisp values, DMUs with fuzzy inputs and outputs were unable to be measured. Due to Kao and Liu’s fuzzy efficiency measures in DEA, Hwang et al.’s is modified into fuzzy DEA to measure DMUs. In fuzzy DEA, DMUs composed of triangular fuzzy numbers are easily yielded their fuzzy efficiency values and then the efficiency values of fuzzy DMUs are rationally judged by corresponding judgement rules. With fuzzy DEA, efficient DMUs are clearly discriminated from inefficient ones, and then efficient DMUs are evaluated by fuzzy technique for order preference by similarity to ideal solution (fuzzy TOPSIS) to find the optimal DMU. According to fuzzy DEA and TOPSIS, the power consumption efficiency of an electromechanical system within a long-term period for sustainability are rationally measured and evaluated.

Appendices

Appendices

Appendix 1

Charnes et al. (Charnes et al. 1978) proposed that the output-oriented CCR model for ${\text{DMU}}{}_{k}$ was presented by a multiplier form as

$$ \begin{gathered} {\text{min}}\frac{{\sum\nolimits_{i = 1}^{m} {v_{i} } x_{ik} }}{{\sum\nolimits_{r = 1}^{s} {u_{r} } y_{rk} }} \hfill \\ {\text{s.t.}}\frac{{\sum\nolimits_{i = 1}^{m} {v_{i} } x_{ij} }}{{\sum\nolimits_{r = 1}^{s} {u_{r} } y_{rj} }} \ge 1\;j = 1,2,...,n \hfill \\ u{}_{r},v_{i} \ge 0,\;i = 1,2,...,m;\;r = 1,2,...,s. \hfill \\ \end{gathered} $$

Since Charnes et al.’s CCR model was fractional programming, measuring a DMU’s efficiency by the above model was complex. Thus, Charnes et al. transformed the fractional programming model into a linear programming one as

$$ \begin{gathered} {\text{min}}\sum\limits_{i = 1}^{m} {v_{i} } x{}_{ik} \hfill \\ {\text{s.t.}}\;\sum\limits_{r = 1}^{s} {u{}_{r}} y{}_{rk} = {1} \hfill \\ \sum\limits_{i = 1}^{m} {v_{i} } x{}_{ij} - \sum\limits_{r = 1}^{s} {u{}_{r}} y{}_{rj} \ge 0\;j = 1,2,...,n \hfill \\ u{}_{r},v_{i} \ge {0}, \;i = 1,2,...,m;\;r = 1,2,...,s. \hfill \\ \end{gathered} $$

Moreover, the dual (i.e., envelope form) of linear programming CCR (i.e., multiplier form) above was shown as:

max $\beta$

$$ \begin{gathered} {\text{s.t.}}\;\sum\limits_{j = 1}^{n} {\lambda {}_{j}} x{}_{ij} \le x{}_{ik}\;i = 1,2,...,m \hfill \\ \sum\limits_{j = 1}^{n} {\lambda {}_{j}} y{}_{rj} \ge \beta y{}_{rk}\;r = 1,2,...,s \hfill \\ \lambda {}_{j} \ge 0,\;j = 1,2,...,n. \hfill \\ \end{gathered} $$

Appendix 2

Bankeret al. (1984) modified Charnes et al.’s output-oriented CCR model into BCC model presented by a multiplier form as

$$ \begin{gathered} {\text{max}}\frac{{\sum\nolimits_{i = 1}^{m} {v_{i} } x_{ik} + v_{0} }}{{\sum\nolimits_{r = 1}^{s} {u_{r} } y_{rk} }} \hfill \\ {\text{s.t.}}\;\frac{{\sum\nolimits_{i = 1}^{m} {v_{i} } x_{ij} + v_{0} }}{{\sum\nolimits_{r = 1}^{s} {u_{r} } y_{rj} }} \ge 1\;j = 1,2,...,n \hfill \\ u{}_{r},v_{i} \ge 0,\;i = 1,2,...,m;\;r = 1,2,...,s. \hfill \\ \end{gathered} $$

The fractional programming model was turned into a linear programming one as

$$ \begin{gathered} {\text{max}}\sum\limits_{i = 1}^{m} {v_{i} } x{}_{ik} + v_{0} \hfill \\ {\text{s.t.}}\;\sum\limits_{r = 1}^{s} {u{}_{r}} y{}_{rk} = 1 \hfill \\ \sum\limits_{i = 1}^{m} {v_{i} } x{}_{ij} - \sum\limits_{r = 1}^{s} {u{}_{r}} y{}_{rj} + v_{0} \ge 0\;j = 1,2,...,n \hfill \\ u{}_{r},v_{i} \ge {0},\;i = 1,2,...,m;\;r = 1,2,...,s. \hfill \\ \end{gathered} $$

The linear programming above had a multiplier form, and its dual model (i.e., envelope form) was displayed as.

$$ \begin{gathered} \max \beta \hfill \\ {\text{s.t.}}\sum\limits_{j = 1}^{n} {\lambda {}_{j}} x{}_{ij} \le x{}_{ik}\;i = 1,2,...,m \hfill \\ \sum\limits_{j = 1}^{n} {\lambda {}_{j}} y{}_{rj} \ge \beta y{}_{rk}\;r = 1,2,...,s \hfill \\ \sum\limits_{j = 1}^{n} {\lambda {}_{j}} = 1 \hfill \\ \lambda {}_{j} \ge 0, \;j = 1,2,...,n. \hfill \\ \end{gathered} $$

Appendix 3

Step 1: Identify a decision matrix for a giving MCDM problem.

The decision matrix is presented below.

$$ C_{1} \quad C_{2} \quad \cdots \quad C_{n} $$

$G = \begin{array}{*{20}c} {A_{1} } \\ {A_{2} } \\ \vdots \\ {A_{m} } \\ \end{array} \left[ {\begin{array}{*{20}c} {G_{11} } & {G_{12} } & \cdots & {G_{1n} } \\ {G_{21} } & {G_{22} } & \cdots & {G_{2n} } \\ \vdots & \vdots & \cdots & \vdots \\ {G_{m1} } & {G_{m2} } & \cdots & {G_{mn} } \\ \end{array} } \right]$ and $W = {(}W_{1} ,W_{2} ,...,W_{n} {)}$, where $A_{1} ,A_{2} ,...,A_{m}$ are feasible alternatives, $C_{1} ,C_{2} ,...,C_{n}$ are evaluation criteria, $G_{ij}$ is the evaluation rating of $A_{i}$ against $C_{j}$, and $W_{j}$ is the weight of $C_{j}$ for $i = 1,2,...,m$; $j = 1,2,...,n$.

Step 2: Normalize the decision matrix.

Let $g_{ij} = \frac{{G_{ij} }}{{\sum\nolimits_{i = 1}^{m} {G_{ij} } }}$ be the normalization of $G_{ij}$ in the decision matrix for $i = 1,2,...,m$; $j = 1,2,...,n$.

Step 3: Construct a weighted decision matrix.

Let $u_{ij} = g_{ij} \times W_{j}$ be the weighted $g_{ij}$ in the weighted decision matrix constructed, where $i = 1,2,...,m$; $j = 1,2,...,n$.

Step 4: Find anti-ideal solution $A^{ - }$ and ideal solution $A^{ + }$.

Let $A^{ - }$$= {(}u_{1}^{ - } ,u_{2}^{ - } ,...,u_{n}^{ - } {)}$ and $A^{ + }$$= {(}u_{1}^{ + } ,u_{2}^{ + } ,...,u_{n}^{ + } {)}$,

where $u_{j}^{ - }$ = $\left\{ {\begin{array}{*{20}c} {\mathop {\min }\limits_{i = 1,2,...,m} \{ u_{ij} \} \, \begin{array}{*{20}c} {if} & {j \in J} \\ \end{array} } \\ {\mathop {\max }\limits_{i = 1,2,...,m} \{ u_{ij} \} \, \begin{array}{*{20}c} {if} & {j \in J} \\ \end{array} ^{\prime}} \\ \end{array} } \right.$, and $u_{j}^{ + }$ = $\left\{ {\begin{array}{*{20}c} {\mathop {\max }\limits_{i = 1,2,...,m} \{ u_{ij} \} \, \begin{array}{*{20}c} {if} & {j \in J} \\ \end{array} } \\ {\mathop {\min }\limits_{i = 1,2,...,m} \{ u_{ij} \} \, \begin{array}{*{20}c} {if} & {j \in J} \\ \end{array} ^{\prime}} \\ \end{array} } \right.$ for $J$ is a set consisting of benefit criteria, and $J^{\prime}$ is a set composed of cost criteria.

Step 5: Yield distances between alternatives and anti-ideal solution/ideal solution.

Let $A_{i}^{ - } = (\sum\nolimits_{j = 1}^{n} {(u_{ij} - u_{j}^{ - } )^{2} } )^{1/2}$ be the distance between alternative $A_{i}$ and anti-ideal solution, $\forall i$.

Let $A_{i}^{ + } = (\sum\nolimits_{j = 1}^{n} {(u_{ij} - u_{j}^{ + } )^{2} )^{1/2} }$ be the distance between alternative $A_{i}$ and ideal solution, $\forall i$.

Step 6: Derive relative closeness coefficients of all alternatives.

Let $A_{i}^{*} = \frac{{A_{i}^{ - } }}{{A_{i}^{ - } + A_{i}^{ + } }}$ be the relative closeness coefficient of $A_{i}$, $i = 1,2,...,m$.

Step 7: Rank alternatives according to their relative closeness coefficients of alternatives.

Obviously, $0 \le A_{i}^{*} \le 1$ and the optimal one in all feasible alternatives has the maximum relative closeness coefficient (Hwang and Yoon 1981).